Dimension Estimates for Badly Approximable Affine Forms

Taehyeong Kim, Wooyeon Kim, Seonhee Lim

correctmedium confidence

Category: math.DS
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper explicitly proves the two-way equivalence in all dimensions (with weights) between: (1) existence of ε>0 such that Bad_A(ε) has full Hausdorff dimension m and (2) A being singular on average. This is stated as Theorem 1.3 and shown by combining Theorem 1.2 for (1)⇒(2) and a new argument in Section 6 for (2)⇒(1) (e.g., Proposition 6.8 together with Propositions 6.10–6.11), culminating in the full-dimensionality conclusion; see the statement of Theorem 1.3 and its proof outline in the introduction, and the detailed proof of (2)⇒(1) in Section 6 . By contrast, the model asserts that (2)⇒(1) is known only in 1D and was likely open as of 2021-11-30, which is contradicted by the dated manuscript (Nov 30, 2021) containing the general proof.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript establishes an important equivalence in the general weighted multi-dimensional setting and contributes an effective relative-entropy framework that will be useful more broadly. The arguments are sound and the structure is clear. Minor issues (small typos and a few clarifications on constants and assumptions) can be addressed quickly.